Models of Peano Arithmetic

Monday, February 27, 2012 7:00 pm Room 4214-03

Professor Erez Shochat (St. Francis College)

Automorphism groups of saturated models of PA

Abstract. In this talk we survey recent results of Nurkhaidarov and

Schmerl concerning the automorphism groups or saturated models of PA.

Computational Logic Seminar

Time 2:00 – 4:00 PM, Room 3209, February 28

Speaker: Melvin Fitting (CUNY)

Title: Possible World Semantics for First Order LP

Abstract: In work of Artemov and Yavorskaya an elegant formulation of

the first-order logic of proofs was given, FOLP. This logic plays a

fundamental role in providing an arithmetic semantics for first-order

intuitionistic logic, as they showed. In particular, they proved an

arithmetic soundness theorem, and a realization theorem for FOLP.

Continuing with the study of FOLP, we provide a possible-world

semantics, based on our earlier propositional semantics for LP. We

also give an Mkrtychev semantics. We will present the semantics in

detail, but only sketch the completeness argument itself, since it is

a straightforward extension of familiar methods.

Set Theory Seminar

Friday, March 2, 2012 10:00 am GC 6417

Professor Dimitris Vlitas

An Infinite self dual Theorem

Abstract. Recall that the classical Ramsey theorem states that given

any finite coloring of the set of all K elements subsets of \omega

there exists of an infinite subset A \subseteq \omega where the

restriction of the coloring is constant. The dual form of Ramsey

theorem, the Carlson-Simpson Theorem, states that given any finite

Borel coloring of the set of all partitions of \omega into K many

classes, there exists a partition r of \omega into \omega many classes

such that the set of all K partitions of \omega resulting by

identifying classes of r is monochromatic. There are also the

corresponding finite versions of these results, the finite Ramsey

Theorem, and the Graham-Rothschild theorem, respectively. S. Solecki

recently proved a self dual theorem that implies simultaneously the

finite version of the Ramsey theorem and the Graham-Rothschild

theorem. He achieved that by introducing the notion of a connection,

which roughly speaking is a labelled partition of L into K many

classes, for K and L integers. He then proved that given any positive

integers K,L and M there exists N such that for any L coloring of all

labelled partitions of N into K many pieces, there exists a labelled

partition of M into K pieces, such that the set of all labelled

partitions of N into M composed with the particular labelled partition

of M into K is monochromatic. The composition is defined in the most

natural way by composing partitions, namely that partition N into M

pieces and then M into K pieces, so we finally partition N into K. The

composition of the label functions is done in the reverse order. We

extend canonically his notion of connection to labeled partitions of

\omega, with finite or infinitely many classes and we prove the

following: For any finite Borel coloring of all labelled K-partitions

of \omega there is a fixed labelled \omega-partition of \omega such

that the set of all of its reductions, ie labelled K-partitions of

\omega which result from putting pieces of the fixed partition

together, is monochromatic. The proof is done by induction on K and

the use of the left variable Hales-Jewett Theorem. In the final

section of the paper we extend this result by building the

corresponding topological Ramsey space F_{\omega,\omega}.

Model Theory Seminar

Friday, March 2, 2012 12:30 pm GC 6417

Professor Jouko Väänänen (University of Helsinki and University of Amsterdam)

On second-order model theory II

Abstract. This mini course will cover the following topics: second

order characterizable structures, Henkin models, Completeness Theorem,

internal categoricity, and existence of second order equivalent

non-isomorphic models.

Logic Workshop + Model Theory, Number Theory, Algebraic Geometry Umbrella series

Friday, March 2, 2012 2:00 pm GC 6417

Professor Zoe Chatzidakis (Université Paris 7)

Algebraic dynamics, difference fields and model theory

Abstract. In this talk I will discuss how to obtain the following result:

Let K_1, K_2 be algebraically closed fields intersecting in K (and

contained in some large algebraically closed field Omega), and for

i=1,2, let V_i be a variety defined over K_i, and f_i a dominant

rational map of V_i to V_i, with deg(f_2)>1. Viewing the V_i’s as

varieties defined over some algebraically closed L containing K_1 and

K_2, assume that there is a dominant rational map h from V_1 to V_2,

such that f_2h=hf_1.

Then there is a variety V defined over K, a dominant rational map f

from V to V, and a dominant rational map g from V_2 to V such that

fg=gf_2, and with deg(f)=deg(f_2).

This can be summarized as follows: if the algebraic dynamics (V_1,f_1)

defined over K_1 dominates the algebraic dynamics (V_2,f_2) defined

over K_2 and with deg(f_2)>1, then (V_2,f_2) dominates some (V,f)

defined over K and with deg(f)=deg(f_2).

Some precisions on the terminology: my varieties are irreducible,

affine or projective. An interesting case is when K_1=K.

The mysterious model-theoretic result from which the result follows,

is the following:

Let T be a theory which is supersimple and has the CBP. Let K_1 and

K_2 be algebraically closed subsets of a model M, intersecting in K,

a_1, a_2 tuples in M.

Assume that a_1 is independent from K_2 over K_1, a_2 is independent

from K_1 over K_2 and of finite rank over K_2. Assume furthermore that

tp(K_2/K) is almost p-internal, p a type of rank 1 over K, and that

a_2 is in acl(K_1K_2a_1). Then there is a_3 in dcl(K_2 a_2) which is

independent from K_2 over K, and such that tp(a_2/Ka_3) is almost

p-internal.

I will not explain the hypotheses on T, nor the proof of this result.

Let me just say that the theory of any existentially closed difference

field (Omega,\sigma) satisfy the hypotheses. I will explain how,

taking p to be a 1-type containing \sigma(x)=x, and translating the

algebraic dynamics problem into a problem about difference fields, one

obtains the result.

Recall that tp(K_2/K) is almost p-internal means that there is some L

independent from K_2 over K, and a set b of realisations of p, such

that K_2 is contained in acl(Lb).

When deg(f_2)=1, a similar result holds: (V_2,f_2) has a quotient

(V,f) defined over K (with dim(V)>1), but the proof is completely

different. I will not speak about this case (Jt with E. Hrushovski).

This is a joint talk with the CUNY Logic Workshop

Next Week in Logic at CUNY:

– – – – Monday, Mar 5, 2012 – – – –

– – – – Tuesday, Mar 6, 2012 – – – –

Computational Logic Seminar

Time 2:00 – 4:00 PM, Room 3209, March 6

Speaker: Cagil Tasdemir (Graduate Center)

Title: (provisional) Distributed Knowledge Authorization Language.

– – – – Wednesday, Mar 7, 2012 – – – –

Model Theory, Number Theory, Algebraic Geometry Umbrella series

Wednesday, March 7, 2012 4:30 pm Room C201

MONTAGU / Spring2012 + Model Theory Seminar / Spring2012

Dr. Ivan Tomasić (Queen Mary, University of London)

A twisted theorem of Chebotarev

Abstract. We will discuss a function field version of a theorem of

Chebotarev describing the distribution of local Frobenius elements

associated with a Galois covering of difference-algebraic varieties.

The result is stated in a shape of a trace formula, and one crucial

ingredient is Hrushovski’s twisted Lang-Weil estimate for the number

of points of a difference variety over the algebraic closure of a

finite field with a power of Frobenius. We will also give a general

description of definable sets over fields with Frobenius in terms of

twisted Galois covers.

MONTAGU is an interdisciplinary seminar organized by Roman Kossak,

Philipp Rothmaler, Hans Schoutens, and Lucien Szpiro

– – – – Thursday, Mar 8, 2012 – – – –

– – – – Friday, Mar 9, 2012 – – – –

CONFERENCE: Spring 2012 Mid-Atlantic Mathematical Logic Seminar

For talk abstracts and further details visit the website:

http://nylogic.org/MAMLS_GC_2012/MAMLS_GC_2012/MAMLS_GC_2012_talks

Friday, March 9, 2012 1:00 pm GC 4102 (Science Center)

Professor Mariya Soskova (Sofia University Faculty of Mathematics

and Informatics)

Extensions of the Turing model for relative definability

Friday, March 9, 2012 2:00 pm GC 4102 (Science Center)

Dr. David Aspero (University of Vienna, Insitute for Logic)

Wishful thinking in set theory: Omega-completeness and maximality

Friday, March 9, 2012 3:30 pm GC 4102 (Science Center)

Professor Jouko Väänänen (University of Helsinki and University of Amsterdam)

Sort Logic and Foundations of Mathematics

Friday, March 9, 2012 4:45 pm GC 4102 (Science Center)

Panel discussion: the unity and diversity of logic

The panelists will be Prof. Gregory Cherlin, Prof. Joel David Hamkins,

Prof. Rohit Parikh, and Prof. Jouko Väänänen, with the discussion

moderated by Prof. Russell Miller. Questions and participation from

the audience are encouraged.

– – – – Friday, Mar 9, 2012 – – – –

CONFERENCE: Spring 2012 Mid-Atlantic Mathematical Logic Seminar

For talk abstracts and further details visit the website:

http://nylogic.org/MAMLS_GC_2012/MAMLS_GC_2012/MAMLS_GC_2012_talks

Saturday, March 10, 2012 10:00 am Baruch College – Room 4120 VC

Professor Jan Reimann (Pennsylvania State University)

Degree-invariant functions and uniform Borel reductions

Saturday, March 10, 2012 11:00 am Baruch College – Room 4120 VC

Professor Henry Towsner (University of Connecticut)

Extracting information from proofs of sentences with set quantifiers

Saturday, March 10, 2012 2:00 pm Baruch College – Room 4120 VC

Professor Juliette Kennedy (University of Helsinki)

Change the logic, change the meaning? Quine’s dictum, formalism

freeness, and inner models from extended logics

Saturday, March 10, 2012 3:00 pm Baruch College – Room 4120 VC

Professor Karen Lange (Wellesley College)

Degrees of orderings on torsion-free abelian groups

Saturday, March 10, 2012 4:30 pm Baruch College – Room 4120 VC

Professor Gregory Cherlin (Rutgers University)

Metrically homogeneous graphs

– – – – Other Logic News – – – –

Introducing MONTAGU, the Model Theory, Number Theory, Algebraic

Geometry Umbrella series

http://nylogic.org/MONTAGU

The Model Theory, Number Theory and Algebraic Geometry Umbrella

seminar (aka MONTAGU) is a newly launched collaborative seminar which

will feature special, interdisciplinary events, connecting

model-theory and logic with algebraic and arithmetic number theory.

Events will be sponsored by the Model Theory Seminar, the Commutative

Algebra and Algebraic Geometry (CAAG) seminar and the Joint Number

Theory Seminar. This seminar is organized by Roman Kossak, Philipp

Rothmaler, Hans Schoutens, and Lucien Szpiro.

See events Friday, March 2nd and Wednesday, March 7th.

– – – – Web Site – – – –

The majority of this information, including an interactive calendar of

future events, can be found at our website:

http://nylogic.org/Calendar

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